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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
Thursday at 11:16 PM
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

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[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies
jlacosta
Thursday at 11:16 PM
0 replies
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k i Adding contests to the Contest Collections
dcouchman   1
N Apr 5, 2023 by v_Enhance
Want to help AoPS remain a valuable Olympiad resource? Help us add contests to AoPS's Contest Collections.

Find instructions and a list of contests to add here: https://artofproblemsolving.com/community/c40244h1064480_contests_to_add
1 reply
dcouchman
Sep 9, 2019
v_Enhance
Apr 5, 2023
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k i Zero tolerance
ZetaX   49
N May 4, 2019 by NoDealsHere
Source: Use your common sense! (enough is enough)
Some users don't want to learn, some other simply ignore advises.
But please follow the following guideline:


To make it short: ALWAYS USE YOUR COMMON SENSE IF POSTING!
If you don't have common sense, don't post.


More specifically:

For new threads:


a) Good, meaningful title:
The title has to say what the problem is about in best way possible.
If that title occured already, it's definitely bad. And contest names aren't good either.
That's in fact a requirement for being able to search old problems.

Examples:
Bad titles:
- "Hard"/"Medium"/"Easy" (if you find it so cool how hard/easy it is, tell it in the post and use a title that tells us the problem)
- "Number Theory" (hey guy, guess why this forum's named that way¿ and is it the only such problem on earth¿)
- "Fibonacci" (there are millions of Fibonacci problems out there, all posted and named the same...)
- "Chinese TST 2003" (does this say anything about the problem¿)
Good titles:
- "On divisors of a³+2b³+4c³-6abc"
- "Number of solutions to x²+y²=6z²"
- "Fibonacci numbers are never squares"


b) Use search function:
Before posting a "new" problem spend at least two, better five, minutes to look if this problem was posted before. If it was, don't repost it. If you have anything important to say on topic, post it in one of the older threads.
If the thread is locked cause of this, use search function.

Update (by Amir Hossein). The best way to search for two keywords in AoPS is to input
[code]+"first keyword" +"second keyword"[/code]
so that any post containing both strings "first word" and "second form".


c) Good problem statement:
Some recent really bad post was:
[quote]$lim_{n\to 1}^{+\infty}\frac{1}{n}-lnn$[/quote]
It contains no question and no answer.
If you do this, too, you are on the best way to get your thread deleted. Write everything clearly, define where your variables come from (and define the "natural" numbers if used). Additionally read your post at least twice before submitting. After you sent it, read it again and use the Edit-Button if necessary to correct errors.


For answers to already existing threads:


d) Of any interest and with content:
Don't post things that are more trivial than completely obvious. For example, if the question is to solve $x^{3}+y^{3}=z^{3}$, do not answer with "$x=y=z=0$ is a solution" only. Either you post any kind of proof or at least something unexpected (like "$x=1337, y=481, z=42$ is the smallest solution). Someone that does not see that $x=y=z=0$ is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
Similar, posting "I have solved this problem" but not posting anything else is not welcome; it even looks that you just want to show off what a genius you are.

e) Well written and checked answers:
Like c) for new threads, check your solutions at least twice for mistakes. And after sending, read it again and use the Edit-Button if necessary to correct errors.



To repeat it: ALWAYS USE YOUR COMMON SENSE IF POSTING!


Everything definitely out of range of common sense will be locked or deleted (exept for new users having less than about 42 posts, they are newbies and need/get some time to learn).

The above rules will be applied from next monday (5. march of 2007).
Feel free to discuss on this here.
49 replies
ZetaX
Feb 27, 2007
NoDealsHere
May 4, 2019
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D1023 : MVT 2.0
Dattier   1
N a few seconds ago by Dattier
Source: les dattes à Dattier
Let $f \in C(\mathbb R)$ derivable on $\mathbb R$ with $$\forall x \in \mathbb R,\forall h \geq 0, f(x)-3f(x+h)+3f(x+2h)-f(x+3h) \geq 0$$
Is it true that $$\forall (a,b) \in\mathbb R^2, |f(a)-f(b)|\leq \max\left(\left|f'\left(\dfrac{a+b} 2\right)\right|,\dfrac {|f'(a)+f'(b)|}{2}\right)\times |a-b|$$
1 reply
Dattier
Apr 29, 2025
Dattier
a few seconds ago
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Equivalent condition of the uniformly continuous fo a function
Alphaamss   0
21 minutes ago
Source: Personal
Let $f_{a,b}(x)=x^a\cos(x^b),x\in(0,\infty)$. Get all the $(a,b)\in\mathbb R^2$ such that $f_{a,b}$ is uniformly continuous on $(0,\infty)$.
0 replies
Alphaamss
21 minutes ago
0 replies
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Find all continuous functions
bakkune   2
N 34 minutes ago by bakkune
Source: Own
Find all continuous function $f, g\colon\mathbb{R}\to\mathbb{R}$ satisfied
$$ (x - k)f(x) = \int_k^x g(y)\mathrm{d}y $$for all $x\in\mathbb{R}$ and all $k\in\mathbb{Z}$.
2 replies
bakkune
2 hours ago
bakkune
34 minutes ago
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ISI 2019 : Problem #2
integrated_JRC   40
N an hour ago by Sammy27
Source: I.S.I. 2019
Let $f:(0,\infty)\to\mathbb{R}$ be defined by $$f(x)=\lim_{n\to\infty}\cos^n\bigg(\frac{1}{n^x}\bigg)$$(a) Show that $f$ has exactly one point of discontinuity.
(b) Evaluate $f$ at its point of discontinuity.
40 replies
integrated_JRC
May 5, 2019
Sammy27
an hour ago
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Equal sum of digits
Fudicuehfosonrcjeong   0
an hour ago
Is it true that for any two positive integers a, b there exists a positive integer k such that s(ka)=s(kb), where s(n) is sum of digits in base 10?
0 replies
Fudicuehfosonrcjeong
an hour ago
0 replies
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Common tangent of mixtilinear incircles
CyclicISLscelesTrapezoid   3
N 2 hours ago by Ilikeminecraft
Source: MOP 2020/1Z
Let $ABCD$ be a quadrilateral inscribed in circle $\Omega$. Circles $\omega_A$ and $\omega_D$ are drawn internally tangent to $\Omega$, such that $\omega_A$ is tangent to $\overline{AB}$ and $\overline{AC}$ while $\omega_D$ is tangent to $\overline{DB}$ and $\overline{DC}$. Prove that we can draw a line parallel to $\overline{AD}$ which is simultaneously tangent to both $\omega_A$ and $\omega_D$.
3 replies
CyclicISLscelesTrapezoid
Jan 6, 2023
Ilikeminecraft
2 hours ago
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Construct
Pomegranat   2
N 2 hours ago by Blackbeam999
Source: idk
Let \( p \) be a prime number. Prove that there exists a natural number \( n \) such that
\[ p \mid m^n - n. \]
2 replies
Pomegranat
Apr 30, 2025
Blackbeam999
2 hours ago
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square root problem
kjhgyuio   2
N 2 hours ago by wh0nix
........
2 replies
kjhgyuio
3 hours ago
wh0nix
2 hours ago
J
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Almost Squarefree Integers
oVlad   4
N 2 hours ago by HeshTarg
Source: Romania Junior TST 2025 Day 1 P1
A positive integer $n\geqslant 3$ is almost squarefree if there exists a prime number $p\equiv 1\bmod 3$ such that $p^2\mid n$ and $n/p$ is squarefree. Prove that for any almost squarefree positive integer $n$ the ratio $2\sigma(n)/d(n)$ is an integer.
4 replies
oVlad
Apr 12, 2025
HeshTarg
2 hours ago
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A nice and easy gem off of StackExchange
NamelyOrange   2
N 3 hours ago by Royal_mhyasd
Source: https://math.stackexchange.com/questions/3818796/
Define $S$ as the set of all numbers of the form $2^i5^j$ for some nonnegative $i$ and $j$. Find (with proof) all pairs $(m,n)$ such that $m,n\in S$ and $m-n=1$.


Rephrased: Solve $2^a5^b-2^c5^d=1$ over $(\mathbb{N}_0)^4$, and prove that your solution(s) is/are the only one(s).
2 replies
1 viewing
NamelyOrange
Yesterday at 8:13 PM
Royal_mhyasd
3 hours ago
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Comics and triangles in perspective
srirampanchapakesan   1
N 3 hours ago by ohiorizzler1434
Source: Own
Let a conic intersect the sides BC, CA, AB of triangle ABC at A1,A2,B1,B2,C1,C2.

T1 is the triangle formed by A1B2, B1C2, and C1A2.

T2 is the triangle formed by A2B1, B2C1 and C2A1.

Prove that the triangles ABC, T1 and T2 are pair-wise in perspective.

Also prove that all three centers of perspective coincide.
1 reply
srirampanchapakesan
4 hours ago
ohiorizzler1434
3 hours ago
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Integer a_k such that b - a^n_k is divisible by k
orl   70
N 3 hours ago by Aiden-1089
Source: IMO Shortlist 2007, N2, Ukrainian TST 2008 Problem 10
Let $b,n > 1$ be integers. Suppose that for each $k > 1$ there exists an integer $a_k$ such that $b - a^n_k$ is divisible by $k$. Prove that $b = A^n$ for some integer $A$.

Author: Dan Brown, Canada
70 replies
orl
Jul 13, 2008
Aiden-1089
3 hours ago
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minimum of \sqrt{\frac{a}{b(3a+2)}}+\sqrt{\frac{b}{a(3b+2)}}
parmenides51   10
N 3 hours ago by Tomilovedoingmath
Source: JBMO Shortlist 2017 A2
Let $a$ and $b$ be positive real numbers such that $3a^2 + 2b^2 = 3a + 2b$. Find the minimum value of $A =\sqrt{\frac{a}{b(3a+2)}} + \sqrt{\frac{b}{a(2b+3)}} $
10 replies
parmenides51
Jul 25, 2018
Tomilovedoingmath
3 hours ago
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inequalities
Cobedangiu   1
N 4 hours ago by xytunghoanh
$a,b,c>0$ and $\sum ab=\dfrac{1}{3}$. Prove that:
$\sum \dfrac{1}{a^2-bc+1}\le 3$
1 reply
Cobedangiu
4 hours ago
xytunghoanh
4 hours ago
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Integrable function: + and - on every subinterval.
SPQ   3
N Apr 23, 2025 by solyaris
Provide a function integrable on [a, b] such that f takes on positive and negative values on every subinterval (c, d) of [a, b]. Prove your function satisfies both conditions.
3 replies
SPQ
Apr 23, 2025
solyaris
Apr 23, 2025
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Integrable function: + and - on every subinterval.
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Reveal topic tags
function
real analysis
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SPQ
5 posts
SPQ
#1 h  Apr 23, 2025, 2:40 AM
Y by
Provide a function integrable on [a, b] such that f takes on positive and negative values on every subinterval (c, d) of [a, b]. Prove your function satisfies both conditions.
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greenturtle3141
3557 posts
greenturtle3141
#2 h  Apr 23, 2025, 4:26 AM
Y by
Really depends what you mean by "integrable", but for the lebesgue case the characteristic function on rationals works just fine
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SPQ
5 posts
SPQ
#3 h  Apr 23, 2025, 5:07 AM
Y by
Thanks for the suggestion. You are right that the characteristic function of the rationals in [a, b] is Lebesgue integrable. However, the characteristic function doesn't satisfy the second condition: it doesn't take on negative values anywhere; it only takes on the values 1 and 0.

Also, just to clarify, I was referring to Riemann integrability.
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solyaris
632 posts
solyaris
#4 h  Apr 23, 2025, 7:06 AM
Y by
@above: But of course you can slightly modify the charactersistic function (so that it takes values $1,-1$ instead of $1,0$) to give a counterexample.

For Riemann integrability you only have to introduce a further slight modification: Take $f(x) = 0$ for irrational $x$, and for rational $x = \frac k n$
with integer $k,n$ and $n > 0$ in reduced form, define $f(\frac k n) = -\frac 1 n$ for odd $n$ and $f(\frac k n) = \frac 1 n$ for even $n$. It is not hard to see that $f$ is Riemann-integrable and takes positive and negative values in any open interval.
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